Drag equation

In fluid dynamics, the drag equation is a formula used to calculate the force of drag experienced by an object due to movement through a fully enclosing fluid. The equation is: $${\displaystyle F_{\rm {d}}\,=\,{\tfrac {1}{2}}\,\rho \,u^{2}\,c_{\rm {d}}\,A}$$ where

• ${\displaystyle F_{\rm {d}}}$ is the drag force, which is by definition the force component in the direction of the flow velocity,
• ${\displaystyle \rho }$ is the mass density of the fluid,^[1]
• ${\displaystyle u}$ is the flow velocity relative to the object,
• ${\displaystyle A}$ is the reference area, and
• ${\displaystyle c_{\rm {d}}}$ is the drag coefficient – a dimensionless coefficient related to the object's geometry and taking into account both skin friction and form drag. If the fluid is a liquid, ${\displaystyle c_{\rm {d}}}$ depends on the Reynolds number; if the fluid is a gas, ${\displaystyle c_{\rm {d}}}$ depends on both the Reynolds number and the Mach number.

The equation is attributed to Lord Rayleigh, who originally used L² in place of A (with L being some linear dimension).^[2]

The reference area A is typically defined as the area of the orthographic projection of the object on a plane perpendicular to the direction of motion. For non-hollow objects with simple shape, such as a sphere, this is exactly the same as the maximal cross sectional area. For other objects (for instance, a rolling tube or the body of a cyclist), A may be significantly larger than the area of any cross section along any plane perpendicular to the direction of motion. Airfoils use the square of the chord length as the reference area; since airfoil chords are usually defined with a length of 1, the reference area is also 1. Aircraft use the wing area (or rotor-blade area) as the reference area, which makes for an easy comparison to lift. Airships and bodies of revolution use the volumetric coefficient of drag, in which the reference area is the square of the cube root of the airship's volume. Sometimes different reference areas are given for the same object in which case a drag coefficient corresponding to each of these different areas must be given.

For sharp-cornered bluff bodies, like square cylinders and plates held transverse to the flow direction, this equation is applicable with the drag coefficient as a constant value when the Reynolds number is greater than 1000.^[3] For smooth bodies, like a cylinder, the drag coefficient may vary significantly until Reynolds numbers up to 10⁷ (ten million).^[4]